-2x+y=15text{and}3x+2y=-33 eching

$$- 2 x + y = 15 \text { and } 3 x + 2 y = - 33$$

x, y uchun yechish

$$x=-9$$

$$y=-3$$

O'rnini bosishdan foydalanish qoidalari

Almashtirishdan foydalanib tenglamalar juftligini yechish uchun, avval o'zgaruvchan qiymatlardan biri uchun tenglamani yeching. So'ngra ana shu o'zgaruvchan natijani boshqa tenglama bilan almashtiring.

$$-2x+y=15,3x+2y=-33$$

Tenglamalardan birini tanlang va teng belgisining chap tomonidagi x ni izolyatsiyalash orqali x ni hisoblang.

$$-2x+y=15$$

Tenglamaning ikkala tarafidan y ni ayirish.

$$-2x=-y+15$$

Ikki tarafini -2 ga bo‘ling.

$$x=-\frac{1}{2}\left(-y+15\right)$$

-\frac{1}{2} ni -y+15 marotabaga ko'paytirish.

$$x=\frac{1}{2}y-\frac{15}{2}$$

\frac{-15+y}{2} ni x uchun boshqa tenglamada almashtirish, 3x+2y=-33.

$$3\left(\frac{1}{2}y-\frac{15}{2}\right)+2y=-33$$

3 ni \frac{-15+y}{2} marotabaga ko'paytirish.

$$\frac{3}{2}y-\frac{45}{2}+2y=-33$$

\frac{3y}{2} ni 2y ga qo'shish.

$$\frac{7}{2}y-\frac{45}{2}=-33$$

\frac{45}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$\frac{7}{2}y=-\frac{21}{2}$$

Tenglamaning ikki tarafini \frac{7}{2} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$y=-3$$

-3 ni y uchun x=\frac{1}{2}y-\frac{15}{2} da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=\frac{1}{2}\left(-3\right)-\frac{15}{2}$$

\frac{1}{2} ni -3 marotabaga ko'paytirish.

$$x=\frac{-3-15}{2}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{15}{2} ni -\frac{3}{2} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=-9$$

Tizim hal qilindi.

$$x=-9,y=-3$$

Matritsalardan foydalanish qadamlari

Tenglamalar standart shaklda ko'rsatilsin so'ng tenglamalar tizimini yechish uchun matritsalardan foydalanilsin.

$$-2x+y=15,3x+2y=-33$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}-2&1\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\-33\end{matrix}\right)$$

\left(\begin{matrix}-2&1\\3&2\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}-2&1\\3&2\end{matrix}\right))\left(\begin{matrix}-2&1\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&1\\3&2\end{matrix}\right))\left(\begin{matrix}15\\-33\end{matrix}\right)$$

Matritsaning ko‘paytmasi va teskarisi o‘zaro teng matristsadir.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&1\\3&2\end{matrix}\right))\left(\begin{matrix}15\\-33\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&1\\3&2\end{matrix}\right))\left(\begin{matrix}15\\-33\end{matrix}\right)$$

\left(\begin{matrix}a&b\\c&d\end{matrix}\right) 2\times 2 matrix uchun, teskari matritsa \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), shuning uchun matritsa tenglamasini matritsani ko‘paytirish masalasi sifatida qayta yozish mumkin.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{-2\times 2-3}&-\frac{1}{-2\times 2-3}\\-\frac{3}{-2\times 2-3}&-\frac{2}{-2\times 2-3}\end{matrix}\right)\left(\begin{matrix}15\\-33\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{7}&\frac{1}{7}\\\frac{3}{7}&\frac{2}{7}\end{matrix}\right)\left(\begin{matrix}15\\-33\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{7}\times 15+\frac{1}{7}\left(-33\right)\\\frac{3}{7}\times 15+\frac{2}{7}\left(-33\right)\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9\\-3\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=-9,y=-3$$

Chiqarib tashlashdan foydalanish qadamlari

Chiqarib tashlash bilan yechim hosil qilish uchun, o'zgartmalarning koeffitsienti ikkala tenglamada bir xil bo'lib o'zgaruvchan qiymat birining boshqasidan ayirilganda, bekor qilishi lozim.

$$-2x+y=15,3x+2y=-33$$

-2x va 3x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 3 ga va ikkinchining har bir tarafidagi barcha shartlarni -2 ga ko'paytiring.

$$3\left(-2\right)x+3y=3\times 15,-2\times 3x-2\times 2y=-2\left(-33\right)$$

Qisqartirish.

$$-6x+3y=45,-6x-4y=66$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali -6x+3y=45 dan -6x-4y=66 ni ayirish.

$$-6x+6x+3y+4y=45-66$$

-6x ni 6x ga qo'shish. -6x va 6x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$3y+4y=45-66$$

3y ni 4y ga qo'shish.

$$7y=45-66$$

45 ni -66 ga qo'shish.

$$7y=-21$$

Ikki tarafini 7 ga bo‘ling.

$$y=-3$$

-3 ni y uchun 3x+2y=-33 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$3x+2\left(-3\right)=-33$$

2 ni -3 marotabaga ko'paytirish.

$$3x-6=-33$$

6 ni tenglamaning ikkala tarafiga qo'shish.

$$3x=-27$$

Ikki tarafini 3 ga bo‘ling.

$$x=-9$$

Tizim hal qilindi.

$$x=-9,y=-3$$